Mathematics of Computation

Author(s): Annegret Weng.
Journal: Math. Comp. 72 (2003), 435-458.
MSC (2000): Primary 11Y16, 11Y40, 94A60; Secondary 14K22, 14H45
Posted: May 3, 2002
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Abstract: In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation.

1.: A.O.L. Atkin, The number of points on an elliptic curve modulo a prime, unpublished manuscript, 1991.
2.: A.O.L. Atkin and F. Morain, Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-68. MR 93m:11136
3.: J. von zur Gathen and Victor Shoup, Computing Frobenius maps and factoring polynomials, Comput. Complexity 2 (1992), 187-224. MR 94d:12011
4.: P. Gaudry and R. Harley, Counting points on hyperelliptic curves over finite fields, ANTS IV (2000), 313-332.
5.: J.I.. Igusa, Arithmetic variety of moduli of genus two, Ann. of Math. 72 (1960), 612-649. MR 22:5637
6.: D.E. Knuth, The art of computer programming vol.2, seminumerical algorithms, Addison-Weseley, 1981. MR 83i:68003
7.: N. Koblitz, Primality of the number of points on an elliptic curve over a finite field, Pacific J. Math. 131 (1988), 157-165. MR 89h:11023
8.: -, Hyperelliptic cryptosystems, J. Cryptology 1 (1989), 139-150. MR 90k:11165
9.: S. Lang, Introduction to algebraic and abelian functions, 2nd ed., Springer-Verlag, 1982. MR 84m:14032
10.: -, Complex multiplication, Springer-Verlag, 1983. MR 85f:11042
11.: S. Louboutin and R. Okazaki, Determination of all non-normal quartic cm-fields and of all non-abelian normal octic cm-fields with class number one, Acta Arith. (1994), 47-62. MR 95g:11107
12.: J.-F. Mestre, Construction des courbes de genre 2 a partir de leurs modules, Effective Methods in Algebraic Geometry (Castiglioncello, 1990), Prog. Math., Birkhäuser 94 (1991), 313-334. MR 92g:14022
13.: D. Mumford, Tata lecture on theta, vol. 1, Birkhäuser, 1983. MR 85h:14026
14.: -, Tata lecture on theta, vol. 2, Birkhäuser, 1984. MR 86b:14017
15.: R. Okazaki, On evaluation of -functions over real quadratic fields, J. Math. Kyoto Univ. 31-4 (1991), 1125-1153. MR 93b:11154
16.: S. Paulus and A. Stein, Comparing real and imaginary arithmetics for divisor class groups of hyperelliptic curves, ANTS III, LNCS 1423 (1998), 576-591. MR 2000i:11098
17.: E. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, 1989. MR 92b:11074
18.: S. Pohlig and M. Hellmann, An improved algorithm for computing logarithms over and its cryptographic significance, IEEE Trans. Inform. Theory IT-24 (1978), 106-110.
19.: G. Shimura, Abelian varieties with complex multiplication and modular functions, revised ed., Princeton University Press, 1998. MR 99e:11076
20.: J.A. Solinas, Generalized Mersenne numbers, Technical Reports, CACR, Waterloo (1999).
21.: A.-M. Spallek, Kurven vom Geschlecht 2 und ihre Anwendung in Public-Key-Kryptosystemen, Ph.D. thesis, Institut für Experimentelle Mathematik, Universität GH Essen, 1994.
22.: P. van Wamelen, Examples of genus two cm curves defined over the rationals, Math. Comp. 68 (1999), 307-320. MR 99c:11079
23.: X. Wang, 2-dimensional simple factors of ${J}_0({N})$ , Manuscripta Math. 87 (1995), 179-197. MR 96h:11059
24.: H.J. Weber, Hyperelliptic simple factors of ${J}_0({N})$ with dimension at least 3, Experiment. Math. 6 (1997), 273-287. MR 99e:14054

Annegret Weng
Affiliation: Institute for Experimental Mathematics, University of Essen, D-45326 Essen, Germany
Email: weng@exp-math.uni-essen.de

DOI: 10.1090/S0025-5718-02-01422-9
PII: S 0025-5718(02)01422-9
Received by editor(s): January 19, 2001
Received by editor(s) in revised form: March 29, 2001
Posted: May 3, 2002
Additional Notes: This work was supported by the NRW Forschungsverbund Datensicherheit (see \texttt{www.datensicherheit.nrw.de)} and the DFG (Graduiertenkolleg).
Copyright of article: Copyright 2002, American Mathematical Society

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